CL600 Captain reported entering an incorrect altimeter setting during initial approach; resulting in aircraft being 1000 feet below their assigned altitude and flight towards terrain.

Narrative

We began to brief the approach and arrival into ZZZ about 35 minutes from anticipated landing. The FO was the PF. We were unable to receive the ATIS as were still not within range and therefore unable to determine exactly which runway was in use. Based on the wind conditions (230/25 G40) we expected to land on RW XX and briefed accordingly. Once we were able to establish radio reception to receive the ATIS we learned that RW XY was currently in use and RW XX was closed. We re-briefed the new runway of landing and the approach expected. Due to the reported winds we now had to include more details regarding the crosswind and gusty conditions and the potential for windshear that we anticipated on the approach. This new briefing delayed our descent checklist until we were within 15 miles of the airport. We were given progressive descents to lower altitudes and in the process of running the descent checklist we incorrectly set the altimeters to a setting of 30.31 instead of the broadcasted 29.31. As we leveled off at 3000 the Approach Controller stated the MVA in the area but said nothing about our now true altitude of 2000' MSL (due to the incorrect altimeter setting). A few seconds later the controller announced 'low altitude alert...climb and maintain 3000'. Even though we were at an indicated '3000 ft. MSL'; I realized that the altimeters had been set incorrectly putting us 1000 ft lower than indicated. We immediately corrected the altimeters and began to climb to the correct altitude. We were then provided new radar vectors for the ILS XY and completed the approach and landing without any further problem.Cause: Incorrectly setting altimeters and failure to recognize the error during the 'crosscheck'.Suggestions: Slowing the pace of the checklist challenge and response and taking time to visually confirm settings vs. simply stating the expected response.